Multiquark States

Marek Karliner,1 Jonathan L. Rosner,2 and Tomasz Skwarnicki3

1School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel; email: marek@post.tau.ac.il 2Enrico Fermi Institute and Department of Physics, University of Chicago, 5620 S. Ellis Avenue, Chicago, IL 60637, USA; email: [email protected] 3Department of Physics, Syracuse University, Syracuse, NY 13244, USA; e-mail: [email protected]

November 30, 2017

Abstract Why do we see certain types of strongly interacting elementary particles and not others? This question was posed over 50 years ago in the context of the quark model. M. Gell-Mann and G. Zweig proposed that the known mesons were qq¯ and baryons qqq, with quarks known at the time u (“up”), d (“down”), and s (“strange”) having charges (2/3,–1/3,–1/3). Mesons and baryons would then have integral charges. Mesons such as qqq¯q¯ and baryons such as qqqqq¯ would also have integral charges. Why weren’t they seen? They have now been seen, but only with additional heavy quarks and under conditions which tell us a lot about the strong interactions and arXiv:1711.10626v1 [hep-ph] 29 Nov 2017 how they manifest themselves. The present article describes recent progress in our understanding of such “exotic” mesons and baryons.

To be submitted to Annual Review of Nuclear and Particle Science. Contents

1 INTRODUCTION 1 1.1 Nucleons And Their Molecules ...... 2 1.2 Quark Model ...... 2 1.3 Quantum Chromodynamics ...... 3 1.4 QCD Motivated Models ...... 4 1.4.1 Potential Models ...... 4 1.4.2 Diquarks ...... 5 1.4.3 Tightly Bound Multiquark States ...... 5 1.4.4 Hadrocharmonium ...... 5 1.4.5 Molecular States ...... 6 1.4.6 Cusps and Anomalous Triangle Singularities ...... 6

2 LIGHT MULTIQUARK CANDIDATES 6 2.1 Light Meson Multiquark Candidates ...... 6 2.2 Light Baryon Multiquark Candidates ...... 7

3 HEAVY-LIGHT MULTIQUARK CANDIDATES 7 3.1 Heavy-Light Meson Multiquark Candidates ...... 7 3.2 Heavy-Light Baryon Multiquark Candidates ...... 8

4 HEAVY QUARKONIUM-LIKE MULTIQUARK CANDIDATES 8 4.1 Ground rules ...... 8 4.2 The X(3872) State ...... 9 4.3 Other Near-threshold Quarkonium-like Mesons ...... 11 4.4 Anomalous Vector States ...... 13 4.5 Other Exotic Meson Candidates Detected in B Decays ...... 17 4.6 Quarkonium-like Pentaquark Candidates ...... 19

5 BEYOND DETECTED STATES 21

6 SUMMARY AND OUTLOOK 23

1 INTRODUCTION

Why do we see certain types of elementary particles and not others? This question was posed over 50 years ago in the context of the quark model [1]. M. Gell-Mann and G. Zweig proposed that the known mesons were qq¯ and baryons qqq, with quarks known at the time u (“up”), d (“down”), and s (“strange”) having charges (2/3,–1/3,–1/3). Mesons and baryons would then have integral charges. Mesons such as qqq¯q¯ and baryons such as qqqqq¯ would also have integral charges. Why weren’t they seen? They have now been seen, but only with additional heavy quarks and under conditions which tell us a lot about the strong interactions and how they manifest themselves. The present article describes recent progress in our understanding of such “exotic” mesons and baryons.

1 After some introductory words on early multiquark states, the quark model, and quantum chromodynamics (QCD), we discuss light multiquark candidates in Sec. 2, heavy- light multiquark candidates in Sec. 3, and heavy quarkonium-like multiquark candidates in Sec. 4. We treat states beyond those detected at present in Sec. 5, and summarize in Sec. 6.

1.1 Nucleons And Their Molecules The symmetries of the strong interactions have a long history, starting with isotopic spin (isospin) which recognized the similarity of the neutron and proton despite their different charges. An important role in understanding forces which bind multiple neutrons and protons (nucleons) into nuclei is played by the pion, coupling to nucleons in an isospin- invariant way. With exchange of pions and other heavier mesons, it was possible to understand the masses of nuclei, with the deuteron (a neutron-proton bound state) a case in point. To the degree that nucleons in nuclei retain much of their identity, one may think of nuclei as the first “molecules” of elementary particles.

1.2 Quark Model Starting in the late 1940s, initially in cosmic rays but by 1953 also in particle accelerators, a new degree of freedom, known as strangeness, began to be recognized in mesons and baryons [2]. Mesons and baryons could be classified into isospin multiplets with their charges Q, the third component I3 of their isospin I, and their hypercharge Y (a quantum number conserved in their strong production) related by Q = I3 + Y/2. (The hypercharge is related to a quantum number S, for “strangeness,” by Y = S + B, where B is baryon number.) However, in the 1950s it was not yet understood why certain isospin multiplets appeared and not others, and how the observed ones were related to one another. By the early 1960s, it became clear that low-lying baryons included the nucleon isospin doublet (n, p) with Y = 1, an isospin singlet Λ and an isospin triplet Σ−, Σ0, Σ+ with Y = 0, and an isospin doublet Ξ−, Ξ0 with Y = −1. These could be unified into an eight- dimensional representation of the symmetry group SU(3) [3]. The lowest-lying mesons, including the pion and charged and neutral kaons, also could be fit into an eight-fold multiplet along with a predicted meson called the η, soon discovered [4]. Given the spin J = 1/2 and parity P = + of the neutron and proton, their partners in the SU(3) octet were predicted (and eventually observed) to have J P = 1/2+. But by the early 1960s a multiplet of resonant particles with J P = 3/2+ was also taking shape: an isoquartet ∆−, ∆0, ∆+, ∆++ with Y = 1, a heavier isotriplet Σ∗−, Σ∗0, Σ∗+ with Y = 0, and a still heavier isodoublet Ξ∗−, Ξ∗0 with Y = −1. The SU(3) scheme predicted that these were members of a ten-dimensional representation, to be completed by a predicted isosinglet Ω−. It also predicted an equal-spacing rule M(Ω) − M(Ξ∗) = M(Ξ∗) − M(Σ∗) = M(Σ∗) − M(∆). The second equality was known to hold, and the predicted Ω was discovered in 1964, cementing confidence in SU(3) [5]. The quark model [1] (see also [6]) provided an explanation of SU(3), with the quarks forming a fundamental triplet out of which all SU(3) representations could be built. For example, the ten-dimensional baryon representation containing four ∆, three Σ∗, two Ξ∗,

2 and one Ω could be regarded as the totally symmetric qqq combinations of u, d, and s. The dynamics of quarks featured prominently at the 1966 International Conference on High energy Physics in Berkeley. It seemed possible to describe several hundred resonant particles in terms of three quarks for baryons and a quark-antiquark pair for mesons. A nagging question dealt with quark statistics. The ∆++ was seen as a ground state of three u quarks with a total J = 3/2, implying total symmetry in its space × spin wave function. But as a state of fermions, its total wave function should be antisymmetric. The invention of another degree of freedom [7], now called color, in which every qqq state could be totally antisymmetric, solved this problem, and provided a basis for the interaction of quarks with one another through the exchange of gluons. This picture came to be known as quantum chromodynamics, or QCD. QCD also explained why quarks could form only integrally-charged states, with their fractional charges masked by binding to other pairs of quarks or antiquarks. However, the question remained, to this day, why other integrally-charged states such as qqq¯q¯ or qqqqq¯, were not observed. Significant evidence for the reality of quarks came from deep inelastic scattering of elec- trons on protons at the Stanford Linear Accelerator Center [8], recoiling against pointlike objects consistent with quarks. That these objects indeed appeared to have fractional charge was indicated by a comparison of deep inelastic electron scattering with that of neutrinos (see, e.g., [9]). The light quarks u, d, s were eventually joined by heavier ones: c (“charm”) [10], b (“beauty” or “bottom”) [11], and t (“top”) [12]. In contrast to the light quarks, whose properties and effective masses inside mesons and baryons were strongly affected by the QCD interaction, c and b are amenable to approximately nonrelativistic descriptions, as their masses (∼ 1.5 and 5 GeV, respectively) exceed their typical kinetic energies (a few hundred MeV) inside mesons and baryons. (Top quarks form only a fleeting association with other quarks before they decay weakly, having a mass of more than twice that of the W boson.)

1.3 Quantum Chromodynamics The theory of the strong interactions, QCD, was born in a mathematical investigation by Yang and Mills [13] of isotopic spin as a gauge theory. In contrast to electrodynamics, the quanta of a gauged isospin theory carry charges. One consequence of this is a different scale dependence of the interaction strength. The electrodynamic force becomes stronger at short distances (large momentum-transfer scales), while in a Yang-Mills type of theory the force becomes weaker at short distances (“asymptotic freedom”). The relevant calculation, not then understood as signaling asymptotic freedom, first appeared in a theory based on gauged SU(2) symmetry [14]. Asymptotic freedom was noticed by ’t Hooft in 1972 but never published [15]; and calculated in all generality for Yang-Mills (non-Abelian gauge) theories by Gross and Wilczek [16] and Politzer [17]. A gauged SU(3) as the theory of the strong interactions contains the “color” ingredient necessary to understand why the ground states of baryons have quark wave functions that are symmetric in space × spin × flavor, where the last term denotes the quark label u, d, s, . . .. Each quark comes in one of three colors, and a wave function totally

3 antisymmetric in color can be constructed by taking one of each color. The behavior of the strong interaction at long distances is also different from that of the electromagnetic interaction. The gluons, quanta of the strong interactions, interact with one another in such a way that lines of force between a quark and an antiquark bunch up into a tube of essentially constant cross-section area, leading by Gauss’ Law to a constant force at large distances, or a linearly rising potential. (For recent comments on this picture see Refs. [18] and references therein.) When this potential becomes strong enough, a new quark-antiquark pair is created, shielding the color charges of the original pair. Thus quark confinement is an essential consequence of QCD. The strengthening of the QCD coupling at low momentum scales and large distance scales means that perturbation theory is unsuitable in that regime. The leading method for dealing with this behavior is to put spacetime onto a lattice. The limit as the lattice spacing a tends to zero then may be taken. However, the presence of pions, with a very long Compton wavelength, means that the total spatial extent of the lattice has to be large. Coupled with the need to take a → 0, this leads to the requirement of very large lattices, and typically fictional pions which are somewhat heavier than the real ones. The time-dependence of a spatial-lattice state can be described by taking Euclidean time, whereby a dependence e−imt in Minkowski space is converted to e−mτ , where τ ≡ it. −m0τ In the limit τ → ∞, a matrix element will behave as e , where m0 is the mass of the lightest contributing intermediate state. Subtracting off this contribution, one can obtain, with some sacrifice in accuracy, the contribution of the next-lowest intermediate state, and this process can be repeated until statistical limitations set in. Lattice QCD has been very successful in reproducing the masses of known states in- volving u, d, s, c, b quarks. It has also been the leading means for calculating form factors and pseudoscalar meson decay constants, which are more sensitive to wave function details. This is particularly so now that virtual light-quark-antiquark pairs have been taken into account in the unquenched approximation. The most sophisticated calculations even con- sider virtual cc¯ pairs when calculating properties of states containing b quarks. Remaining possible sources of uncertainty include the need for proper treatment of chiral fermions and the use of chiral perturbation theory for extrapolation of calculations involving pions down to their physical mass.

1.4 QCD Motivated Models 1.4.1 Potential Models The discoveries of the charmed and beauty quarks, and their rich cc¯ and b¯b spectra, led to approximate descriptions of their spectra by nonrelativistic potential models [19, 20], including those with relativistic corrections [21]. The short-distance behavior of the interquark potential could be described by a Coulomb-like potential, suitably modified by a logarithmic correction due to asymptotic freedom, while the long-distance behavior was linear in the separation r (see the above description of quark confinement). An interpolation between these two behaviors was provided by a potential logarithmic in r ¯ [22], for which the spacing between QQ levels was independent of the mass mQ of the quark Q, as is nearly the case for cc¯ and b¯b systems.

4 Treating light quarks in bound states as having effective masses of several hundred MeV, and taking into account spin-spin (hyperfine) interactions among them, it was even found possible to bypass many details of potential models, gaining an insight into masses of light mesons and baryons or those containing no more than one heavy quark (c or b). This approach was pioneered in Ref. [23] and applied, for example, to baryons containing b quarks in Refs. [24].

1.4.2 Diquarks A baryon is made of three color-triplet quarks, coupled up to a color singlet using the antisymmetric tensor αβγ, where the indices range from 1 to 3. Each quark pair must then act as a color antitriplet. Under some circumstances it is then useful to consider a baryon as a bound state of a color triplet quark and a color-antisymmetric antitriplet diquark. The color antisymmetry of the diquark requires its space × spin × flavor wave function to be symmetric, where flavor denotes quark identity (u, d, s, . . .). For example, the u and d quarks in the isosinglet baryon Λ are in an S wave (space symmetric) and an isospin zero state (flavor antisymmetric), so they must be in a spin zero state (spin antisymmetric). The spin of the Λ is then carried entirely by the strange quark, consistent with its measured magnetic moment [25]. Some light-quark resonances have been identified as candidates for diquark-antidiquark bound states [26, 27], with the last noting a relation to baryon-antibaryon resonances [28] reminiscent of the original Fermi-Yang model of the pion [29] as a nucleon-antinucleon bound state. The past light-quark pentaquark candidates brought attention to a role diquarks can play in formation of such systems [30]. Even though these candidates did not survive experimental scrutiny (see Sec 2.2), the discussion on the role of diquarks in shaping the structure of ordinary and exotic baryons [31] is very much alive today.

1.4.3 Tightly Bound Multiquark States In addition to the above light-quark resonances, some authors have postulated that new resonances including one or more heavy quarks are candidates for tightly bound diquark- antidiquark states [32, 33, 34]. Thus, the X(3872) first observed decaying to J/ψπ+π− [35] would be interpreted as a bound state of a cu diquark and ac ¯u¯ antidiquark. We shall discuss the merits and drawbacks of this assignment presently.

1.4.4 Hadrocharmonium The resonance X(3872) mentioned above can be regarded as a charmonium state embed- ded in light hadronic matter, called hadrocharmonium in Ref. [36]. This classification is motivated by the observation that multiquark states including a cc¯ pair appear to contain only a single charmonium state, whereas one might expect the wave function to involve a linear combination of several charmonium states in a hadronic molecule or generic multi- quark state.

5 1.4.5 Molecular States The wave functions of many exotic multiquark states such as X(3872) appear to consist, at least in part, of pairs of hadrons each containing one heavy quark. Thus, one can identify X(3872) as a bound or nearly bound state of (D0 = cu¯)(D¯ ∗0 =cu ¯ ) + (charge conjugate), as we shall discuss in Sec. 4. Such assignments are favored if the constituents can be bound via exchange of a light pseudoscalar, such as pion [37, 38, 39] or possibly η [40]. As in the case of the deuteron, pion exchange is not the whole story, but, where permitted, dominates the long-range force.

1.4.6 Cusps and Anomalous Triangle Singularities When a decay process involves three particles in the final state, the proximity of S-wave thresholds in two-body rescattering can lead to behavior which can mimic a resonance while only consisting of a cusp. Kinematic enhancements can also be due to anomalous triangle singularities (for an early manifestation in pion-nucleon scattering see [41]), in which resonance-like behavior is seen when all participants in rescattering approach the mass shell. Triangle singularities and methods to identify true resonances as S-matrix poles have been recently discussed in Refs. [42, 43, 44].

2 LIGHT MULTIQUARK CANDIDATES

2.1 Light Meson Multiquark Candidates

3 1 The P-wave qq¯ states of the three light quarks q = u, d, s consist of P0,1,2 and P1 nonets with positive parity. Here the superscript denotes the quark-spin multiplicity 2Sq + 1, while the subscript denotes the total spin J. The J = 0 states can couple to a pair of pseudoscalar mesons in an S wave, and hence their widths and masses are strongly influenced by these couplings. Indeed, one can regard them as linear combinations of qq¯ and meson-meson states. The latter can be thought of as qqq¯q¯, or tetraquarks.A systematic classification of light J = 0 mesons as tetraquarks was made by Jaffe [26, 27]. The two-pseudoscalar-meson channel strongly affects the production and decay of the nonstrange J = 0 mesons f0(980) (isoscalar) and a0(980) (isovector) [25]. They lie very close to the KK¯ threshold and thus may be thought of, in part, as either KK¯ bound states or tetraquarks containing an ss¯ pair. The f0 is seen to decay predominantly to ππ, but 0 is produced primarily in processes which provide an initial ss¯ pair, such as Bs → J/ψ f0 [45]. Another light-quark system in which meson, rather than quark, degrees of freedom ¯ play an important role is the f1 or a1 system decaying to KKπ with mass around 1420 ∗ ¯ 0 MeV. The Dalitz plot near this total mass shows a0 or f0, K ,K resonances between each final-state pair [46]. The f1(1420) thus may not be a genuine resonance but rather a kinematic effect known as a triangle singularity [47]. Cusp-like behavior in scattering amplitudes near S-wave thresholds for new final states is widespread [48]. For one example, diffractive photoproduction of 3π+3π− exhibits a

6 dip near pp¯ threshold [49]. There may also be a pp¯ resonance or bound state near this mass, but the question is not settled [48].

2.2 Light Baryon Multiquark Candidates The quark model for baryons has been very successful in describing them as qqq states, including those with nonzero internal orbital angular momentum. However, final meson- baryon states (and thus states of qq¯+ qqq) play an important role as well. A case in point is the resonance known as Λ(1405), with J P = 1/2−. It has a history going as far back as the late 1950s [50]; for a recent understanding of its structure see Ref. [51]. Its nature is still being debated, though it is a reasonable candidate for a KN¯ bound state. The quark model predicts three J P = 1/2− isospin-zero baryons: a flavor singlet with quark spin 1/2 and two flavor octets, one with quark spin 1/2 and the other with quark spin 3/2. The Λ(1405) appears to be mainly the flavor singlet, with smaller admixtures of the two octets [52]. Two other states, Λ(1670, 1/2−) and Λ(1800, 1/2−) [25], are the orthogonal mixtures. Couplings to the channels Σπ, NK¯ , and Λη probably play some role in the mixing [53]. A candidate for a K+n resonance called Θ+(1540), whose minimal quark content would besuudd ¯ , was observed in the early 2000s [54]. However, it was not confirmed in further experiments [55] and appears to have been a kinematic effect [56].

3 HEAVY-LIGHT MULTIQUARK CANDIDATES

3.1 Heavy-Light Meson Multiquark Candidates The S-wave states of a charmed quark and a light (u, d, s) antiquark are the pseudoscalar 0 + + 1 ∗0 ∗+ ∗+ 3 mesons D , D , and Ds ( S0) and the vector mesons D , D , and Ds ( S1). The P- wave states naturally divide into those with light-quark total angular momentum j = 1/2 P + + P + + (Jj = 01/2, 11/2) and j = 3/2 (Jj = 13/2, 23/2) [57]. They are predicted to decay to ground-state charmed mesons in the following ways, where P stands for π or K: + + ∗ + ∗ + ∗ 01/2 → DP (L = 0); 11/2 → D P (L = 0); 13/2 → D P (L = 2); 23/2 → (D,D )P (L = 2). The states with j = 3/2 decaying via D-waves are expected to be narrow, and indeed correspond to the observed D1(2420), D2(2460), Ds1(2536), and Ds2(2573) [25]. (Here the subscript denotes total J.) Information is fragmentary on the nonstrange j = 1/2 states + but there exists a broad candidate for the nonstrange 01/2 state with mass M = 2318±29 MeV and width Γ = 267 ± 40 MeV [25]. When both strange and nonstrange candidates for the same (J, j) are seen, the strange candidate is about 115 MeV heavier than the + nonstrange candidate. Thus we would expect a strange 01/2 state around 115 + 2318 = 2433 MeV, above the DK threshold of 2362 MeV. What came as a surprise was the observation by the BaBar Collaboration [58] of + a candidate for the strange 01/2 state at 2317 MeV, more than 100 MeV below na¨ıve 0 expectations and 45 MeV below DK threshold. It was seen instead to decay to Dsπ via 0 an isospin-violating transition. A hint of a strange state at 2460 MeV, decaying to Dsπ γ,

7 + was also seen. Its confirmation [59, 60, 61] supplied a candidate for the strange 11/2 state [62, 63], 40 MeV below D∗K threshold. Proposals for explaining the displacement of Ds0(2317) and Ds1(2460) masses from their expected values included the formation of D(∗)K molecules or bound states [64], the existence of tetraquarks [65, 66, 67, 68], and a realization of chiral symmetry which predicted the observed mass pattern a number of years earlier [69, 70, 71, 72]. The yet-to-be-detected conjectured bottom analogues of Ds0(2317) Ds1(2460) are discussed in Sec. 5.

3.2 Heavy-Light Baryon Multiquark Candidates Threshold effects can involve heavy mesons and light-quark baryons, or heavy baryons and light-quark mesons. An example of the former is a charmed baryon resonance Λc(2940), seen decaying to D0p [73]. The mass was seen to be just below D∗0p threshold, suggesting a bound state or molecular interpretation [74, 75]. Recently the LHCb Collaboration [76] 0 0 − P − has analyzed the D p amplitude in Λb → D pπ and finds a resonance favoring J = 3/2 +3.5 +0.1 +8.0 +5.2 P at a mass of 2944.8−2.5 ± 0.4−4.6 MeV with a width of 27.7−6.8 ± 0.8−10.4 MeV. The J assignment is consistent with an S-wave state of a D∗0 and a proton. Following the alleged discovery of the Θ(1540) pentaquark candidate (see Sec. 2.2) a cuudd¯ state was claimed by the H1 Collaboration at HERA in Hamburg [77], correspond- ing to an effective mass of 3.1 GeV in the D∗±p∓ system. It was not confirmed with further data [78].

4 HEAVY QUARKONIUM-LIKE MULTIQUARK CANDIDATES

4.1 Ground rules In this section we shall discuss states containing two heavy quarks Q which cannot be represented as simple QQ¯ excitations, but which require some admixture of light quarks as well. The notation X will stand for neutral “cryptoexotic” states with likely QQq¯ q¯ content. States in this category with J PC = 1−− which can couple directly to a virtual photon will be denoted Y , while those with a charged light-quark pair (e.g., ud¯) will be ¯ denoted Zc (when the heavy pair is cc¯) or Zb (when the heavy pair is bb). Finally, Pc or ¯ Pb will denote a state such as ccuud¯ or bbuud (“pentaquark”). The spectrum of X, Y , Z states is particularly rich for charmonium. Some controversy exists over the quark content, spin, and parity of many of these states. A useful reference to the experimental literature is contained in Ref. [79]. We shall not discuss in any detail states which we believe to have conventional QQ¯ assignments, concentrating instead on X,Y,Z, and PQ candidates.

8 4.2 The X(3872) State The first evidence for a multiquark state involving cc¯ and light quarks came from the decay B → Kπ+π−J/ψ, in which the π+π−J/ψ system showed a narrow peak around 3872 MeV [35]. It has been confirmed by many other experiments [25], as illustrated in Fig. 1. Its width is less than 1.2 MeV, and its J PC has been established as 1++ [80].

͒ 3872 → J/ψͮͯ ̼ → ͒ 3872 ͅ 30 ̼ͮ → LHCb CDF Belle 4k ͒ 3872 ͒ 3872 → J/ψ 100 ͮͅ 20

ͤͤ̅ → 10 ͒ 3872 ƍ ⋯

Events/1.0MeV 2k Events/2.5 MeV Events/2.5 MeV Events/9.5 0 0 ͤ 100 ̼ → Belle CMS 30 LHCb ͒ 3872 ͒ 3872 ͮͯͅ 14k → ψʚ2͍ʛ ͤͤ → 20 50 ͒ 3872 ƍ ⋯ 10

10k MeV Events/10 Events/3.125 MeV Events/3.125

Events/4.0 MeV Events/4.0 0 0 3.82 3.86 3.90 3.8 3.9 4.0 3.83.9 4.0 M(J/ ψπ +π−) [GeV] M(J/ ψπ +π−) [GeV] M( ψ( nS )γ ) [GeV]

ͤ BaBar BaBar ͮ ̾

∗ͤ 400 10 3872 ͒ 3872 ͅ 3872 ͅ

200 → ͒ 3872 → J/ψ! ͮ 3872 → ̾ ̼ → ͒ ͒ ͒ ̼ Events/10 MeV Events/10 Events/2.0 MeV Events/2.0 0 3.90 3.94 0 3.9 4.0 4.1 Karliner,Rosner,Skwarnicki 2017 M(D *0 D0) [GeV] M(J/ ψω ) [GeV]

Figure 1: Production and decay of the X(3872) state. Detailed figure descriptions can be found in the original references, from which the plots have been adapted: top row left Ref. [80], middle Ref. [81], right [82], middle row left Ref. [83], middle Ref. [84], right Ref. [85], bottom row left Ref. [86], and right Ref. [87].

The mass of X(3872), whose 2016 average [25] is 3871.69 ± 0.17 MeV, is sufficiently close to the threshold for D0D¯ ∗0, namely (1864.83 ± 0.05) + (2006.85 ± 0.05) = (3871.68 ± 0.07) MeV, that one cannot tell whether it is a candidate for a bound state or resonance

9 of D0D¯ ∗0. Clearly, however, the neutral-D channel must play an important role in the makeup of X(3872), as also evidenced by a large fall-apart rate of the X(3872) to D0D¯ ∗0, once the kinematic threshold is exceeded [88, 86, 89] (Fig. 1). The D+D∗− threshold, (1869.59 ± 0.09) + (2010.26 ± 0.05) = (3879.85 ± 0.10) MeV, is sufficiently far from M(X(3872)) that the charged-D channel appears to play a much less important role in its composition. The quark makeup of X(3872) thus should include an important ccu¯ u¯ component. Confirmation of this point is provided by the observation of both X(3872) → ωJ/ψ (ω → π+π−π0) [90, 87] and X(3872) → ρ0J/ψ (ρ0 → π+π−) [91, 80], implying that the X(3872) is a mixture of isospins zero and one [38] (Fig. 1). There also appears to be a cc¯ χc1(2P ) component to the X(3872) wave function, as indicated by the ratio of the radiative decays to γJ/ψ and γψ(2S) [85] (Fig. 1),

B(X(3872) → ψ(2S)γ) R ≡ = 2.46 ± 0.64 ± 0.29 . (1) γ B(X(3872) → J/ψγ)

The measured value is consistent with pure charmonium and a mixture of charmonium and a molecular state, but not with a pure molecular state. Additional rather robust evidence for thecc ¯ component is provided by the relatively large cross section for prompt production of X(3872) in pp¯ [92, 93] and pp collisions [94, 84, 95] (Fig. 1), closely following behavior of the ψ(2S) state. In particular, Ref. [96] uses ALICE data on the production of light nuclei with p 10 √ T . GeV in Pb-Pb collisions at sNN = 2.76 TeV to estimate the expected production cross sections of such nuclei in pp collisions at high pT . Hypertriton, helium-3, and deuteron production cross sections are compared to the CMS results for prompt production of X(3872) [84]. Fig. 1 of Ref. [96] shows that the latter is orders of magnitude larger than the former. Also the dependence of the prompt production of X(3872) on its transverse momentum and pseudo-rapidity, as well as the ratio of the prompt production to the pro- duction in B meson decays, closely follow those of the ψ(2S) charmonium state, pointing to the same production mechanism [93, 95]. The cross section for prompt production of these light nuclei falls rapidly with pT because they are rather large. As soon as pT is bigger than their inverse radius, the probability of forming such weakly bound molecular states becomes very small. The X(3872) binding energy is much smaller than the 2.2 MeV deuteron binding energy. Therefore the spatial extent of the molecular component must be much bigger than the already large deuteron size. We can estimate the inverse size of the X(3872) molecular component using the for- mula 1/r = p2µ|∆E| (2) where µ = 967 MeV is the reduced mass and ∆E is the binding energy. ∆E is at most 0.2 MeV, probably less. This gives 1/r . 20 MeV, corresponding to radius of & 10 fermi, really huge. With such a large radius the cross section for production of the molecular component at pT & 10 GeV is expected to be negligible. Therefore X(3872) must have a significant cc¯ component, whose size is the typical hadronic radius < 1 fermi, much

10 smaller than the size of the molecular component.1 This of course raises the interesting question of how the mixing works for two states whose sizes differ by at least a factor of 10. Perhaps X(3872) lives long enough to make even a small spatial overlap sufficient for significant mixing.2 It is also possible that the molecular component occurs dynamically when the compact X(3872) attempts to disintegrate to D0D¯ ∗0. Hadronic molecules were proposed some time ago [37, 99, 100, 101]. One-pion exchange plays an important (though not exclusive) role in facilitating binding. The attractive force between two states of isospin I1,2 and spin S1,2 transforms as

V ∼ ±I1 · I2 S1 · S2 for (qq, qq¯) interactions, (3) and is expected to bind not only D0D¯ ∗0 + c.c. but many other systems as well, including meson-meson, meson-baryon and baryon-baryon [39]. In particular, there should be an ¯∗ analogue Xb of the X(3872), near BB threshold (10604.8 ± 0.4 MeV for neutral B-s and 10604.5 ± 0.4 MeV for B+B−∗) [102]. Because the thresholds for charged and neutral pairs are so similar, isospin impurity in the Xb is expected to be small, and it should be mostly isoscalar. + − CMS and ATLAS have searched for the decay Xb → Υ(1S)π π [103]. The search in this particular channel was motivated by the seemingly analogous decay X(3872) → J/ψπ+π−. This analogy is misguided, however, because for an isoscalar with J PC = 1++ such a decay is forbidden by G-parity conservation [104]. Thus the null result of these searches does not tell us anything about the existence of Xb. The bottomonium state χb1(3P ) has been recently observed [105]. The Xb state could mix with it and share its decay channels, just as X(3872) is likely a mixture of a DD¯ ∗ molecule and χc1(2P ) [102]. However, the mass difference between the observed χb1(3P ) state and BB¯∗ thresholds is about 93 MeV, which makes a significant mixing unlikely. In fact, the observed χb1(3P ) mass is in excellent agreement with the potential model predictions made over 20 years before its first observation [106], while mixing would have likely affected its mass.

4.3 Other Near-threshold Quarkonium-like Mesons

+ − + − + − + − The cross sections for e e → Υ(1S, 2S, 3S)π π and e e → Υ(1√S)K K were found to be surprisingly large near the peak of the Υ(5S) resonance at s ∼ 10.87 GeV [107]. ¯ One possible explanation of this enhancement was the existence of intermediate bbq1q¯2 states (qi denotes a light quark) decaying to Υ(nS)π or Υ(1S)K [108]. Unusual enhance- + − + − ments were also seen in the cross sections for e e → hb(nP )π π (n = 1, 2), where ¯ hb(nP ) denotes a spin-singlet bb resonance with radial quantum number n, orbital angu- lar momentum L = 1, and total spin J = 1 [109]. These effects were found to be due to two

1It was argued that the short-range structure of the molecular wave function is difficult to predict [97, 98], so large values of Rγ and of prompt production cross-section are not incompatible with the molecular behavior of the wave function at large distances. This, however, does not imply that these experimental observations are natural expectations in the molecular model. We side with the argument 3 that an admixture of charmonium 2 P1 state offers the most natural explanation, and in fact, is not incompatible with the molecular behavior of X(3872) at large distances. 2Alex Bondar, private communication.

11 charged bottomonium-like resonances in Υ(5S) decays [110]. The Υ(nS)π (n = 1, 2, 3) spectra are shown in Fig. 2. The peaks have been named Zb(10610) and Zb(10650). Sim- + − ilar peaks are seen in M(hb(nP )π π )(n = 1, 2). The review of Ref. [79] quotes the average masses as M(Zb(10610)) = 10607.2 ± 2.0 MeV and M(Zb(10650) = 10652.2 ± 1.5 MeV.

ͮ ͯ Υʚ5͍ ʛ → ͔ͮʚͤʛͯʚͤʛ Υʚ5͍ ʛ → ͔   ͔

80 ͔  ͮ  ͮ

ʚ͍  Υʚ3͍ʛ → Belle ͜ →

10k  ͮ ͯ

ʚ2͍ʛ Υʚ5͍ʛ → ͔  40  ͮ

80 ͔

ͮ WS RS  ͮ Events/10 MeV Events/10 Events/4 MeV Events/4 0 + h π ʚ̼ → M( b(2S) ) bkg.

0  Υʚ2͍ʛ → ͜ → 10k 40 ∗  ̼ʛ ʚ1͍ʛ ͮ Events/5 MeV Events/5 40 0 ͮ ͮ ͔

80  ͮ WS RS Events/10 MeV Events/10

Events/5 MeV Events/5 0

+ ʚ̼ → M( hb(1S) π ) Υʚ2͍ʛ → 0  Υʚ1͍ʛ → 30

40 ∗ ̼ ∗ ʛ ͮ

20 MeV Events/5

20 ͤ 0 ͮ 10.6 10.7 10 M(B *B(*) ) [GeV] Events/10 MeV Events/10 Events/10 MeV Events/10 0 0 10.6 10.7 10.6 10.7 M( Υ(nS)π+) [GeV] M( Υ(2S) π0) [GeV] Karliner,Rosner,Skwarnicki 2017

Figure 2: Observations of the Zb(10610) and Zb(10650) states. Detailed figure descriptions can be found in the original references, from which the plots have been adapted: left column Ref. [110], middle column top and middle Ref. [110], bottom Ref. [111], and right column Ref. [112].

The masses of the two peaks are very close to the thresholds for BB¯∗ and B∗B¯∗: (10604.0 ± 0.3) MeV and (10649.3 ± 0.5 MeV, respectively. This suggests that their wave functions should largely consist of the respective S-wave “molecular” components BB¯∗ ∗ ¯∗ ¯∗ and B B [113]. In fact, the Zb(10610) fall-apart rate to BB is large, while there is no

12 ¯∗ ∗ ¯∗ evidence for Zb(10650) → BB , which prefers to decay to B B in spite of the smaller phase-space [112] (Fig. 2). The absence of similar effects just above the BB¯ threshold of (10558.6 ± 0.3) MeV points to an important role of one-pion exchange in the formation of these “molecules”, as a pion cannot couple to the pair of pseudoscalar mesons BB¯ [39]. A counterpart to the Zb system has been observed in exotic charmonium states. The thresholds for [neutral, charged] DD¯ ∗ pairs are [(3871.7 ± 0.1), (3879.8 ± 0.1)] MeV, while the thresholds for [neutral, charged] D∗D¯ ∗ pairs are [(4013.7±0.1), (4020.52±0.1)] MeV. States near both these thresholds, respectively called Zc(3900) and Zc(4020), have been observed in decays of the vector meson candidate Y (4260) (see next section). The Zc(3900) is seen in the ππJ/ψ final state as a peak in M(πJ/ψ) [114, 115, 116, 117] and in the πDD¯ ∗ final state as a peak in M(DD¯ ∗ [118, 119], as illustrated in Fig. 3. Its averaged mass is quoted as (3891.2 ± 3.3) MeV [79]. The Zc(4020) is seen in the ππhc final state as ∗ ¯ ∗ ∗ ¯ ∗ a peak in M(πhc) [120, 121] and in the πD D final state as a peak in M(D D ) [122, 123] (Fig. 3). Its averaged mass is quoted as (4022.9 ± 2.8) MeV [79]. As in the case of the ¯ exotic bottomonium Zc states, the absence of DD peaks is circumstantial evidence in favor of a role for pion exchange in forming molecules of open-flavor pairs. As mentioned earlier, such molecules were anticipated shortly after the discovery of charm [99]. As mentioned in Sec. 1.4, many explanations of these near-threshold Zb and Zc states abound. The close correlation between peaks and thresholds would have to be regarded as a coincidence in potential models. The grouping of multiple quarks in an exotic hadron depends to some extent on their masses; an example is the predominance of QQ color antitriplet diquarks in QQq¯1q¯2 hadrons due to the tight binding of the heavy quarks Q with one another. Explanations based on genuine tetraquarks require the observation of isospin partners; the Zc(3900) may be the charged partner of X(3872), though this interpretation is complicated by the isospin impurity of the latter. Many of the vector states to be described in the next section may admit of a hadrocharmonium explanation [36]. Finally, experience with light-quark systems such as f0(980) and Λ(1405) indicates that resonant vs. cusp behavior may be difficult to sort out when new channels are opening up. Experience with Feshbach resonances [124], also associated with the opening of new channels, may be of help here.

4.4 Anomalous Vector States The direct coupling of quarkonium states with J PC = 1−− to virtual photons has made them particularly easy to observe. The charm and bottom quarks were first observed as 3 ¯ a result of these couplings in the (S-wave) 1 S1 states J/ψ(1S) = cc¯ and Υ(1S) = bb, respectively. Weaker couplings to virtual photons also are possessed by the (D-wave) 3 2S+1 1 D1 states. Here we use the notation n LJ , where n is the radial quantum number, S denotes quark spin, L is represented by S,P,D,F,... for L = 0, 1, 2, 3,..., and J denotes total spin of the state. Candidates for such “conventional” vector quarkonia include the following, where we use the name assigned in Ref. [25] and give the approximate mass in MeV: Charmonium: J/ψ(1S)(3097), ψ(2S)(3686), ψ(1D)(3770), ψ(3S)(4040), ψ(2D)(4160), ψ(4S)(4415); Bottomonium: Υ(1S)(9460), Υ(2S)(10023), Υ(3S)(10355), Υ(4S)(10579), Υ(5S)(10860),

13 ͮ ͯ ͮʚͤʛ ͯʚͤʛ ͮ ͯ ͮʚͤʛ ͯʚͤʛ ͙ ͙ → ͔  ͙ ͙ → ͔  ͮ ͮ ͮ ͮ 80 ͔ → ̈́/  ͔ → ͜ 40 60 20 40 + MeV Events/5 Events/10 MeV Events/10 M(J/ ψπ ) + 0 0 M(h cπ ) ͤ ͤ ͤ ͤ 60 ͔ → ̈́/  40 ͔ → ͜

40 20 20 Events/10 MeV Events/10 Events/10 MeV Events/10 M(J/ 0) M(h π0 ) 0 ψπ 0 c ͮ ∗ͤ ͮ ͮ ∗ ∗ ͮ ͔ → ̾ ̾ 60 ͔ → ʚ̾ ̾ ʛ

40 60 20 Events/2.5 MeV Events/2.5 Events/4 MeV Events/4 0 0 3.7 3.8 3.9 4.0 4.1 3.9 4.0 4.1 M(D*0 D+) [GeV] M(D *D*) [GeV] Karliner,Rosner,Skwarnicki 2017 BES III

Figure 3: Observations of the Zc(3900) and Zc(4020) states. Detailed figure descriptions can be found in the original references, from which the plots have been adapted: left column top Ref. [114], middle Ref. [117], bottom Ref. [119], right column top Ref. [120], middle Ref. [121], and bottom Ref. [122].

14 Υ(6S)(11020). The ratio R ≡ σ(e+e− → hadrons)/σ(e+e− → µ+µ−) as measured by BESIII [125] (Fig. 4) peaks prominently around 4040 MeV, and noticeably just above 4400 MeV, motivating the charmonium 3S and 4S assignments for these peaks. A peak associated with the 2D candidate is less prominent, as befits a D-wave state whose coupling to a virtual photon is suppressed. A prominent feature of R is a steep drop around Ec.m. = 4.2 GeV. The change in R is more than one unit, which could signify the total suppression of charm production (∆R = −4/3). Such a sharp dip is often associated with the opening of a new S-wave channel [48], as in the case of I = 0 ππ scattering near KK¯ threshold. Indeed, the lowest-lying two-body S-wave state into which a cc¯ pair can fragment is ¯ DD1 − c.c. [126], where D1 is a P-wave bound state of a charmed quark and a light (¯u or d¯) antiquark with J P = 1+. The minus sign corresponds to the negative C eigenvalue. 2 The lightest established candidate for D1 has a mass of about 2.42 GeV/c , corresponding to a threshold of 4.285 GeV. The cross sections σ(e+e− → f), where f are specific final states, differ considerably from one another (see the mini-review by Eidelman et al. [79]). For this reason, we briefly describe the apparent resonant activity in each final state. Just as in light-quark spec- troscopy, mixing of quark-model configurations can lead to eigenstates favoring individual channels. ππJ/ψ final state: The cross section for e+e− → π+π−J/ψ, as first seen in the radiative return process e+e− → γπ+π−J/ψ by the BaBar Collaboration [133] and confirmed in several other experiments [79], shows a prominent peak around 4260 MeV. It could be ¯ PC a DD1 state with about 25 MeV of binding energy [134]. Weaker evidence for a J = 1−− state around 4008 MeV presented by the Belle Collaboration [135] has not been confirmed by others. Recently the BESIII Collaboration has reported two new structures in σ(e+e− → π+π−J/ψ): one with a mass of (4222.0 ± 3.4) MeV and a broader one with a mass of (4320 ± 13) MeV [127] (Fig. 4). The first could be identified with a shifted Y (4260), while the second has been proposed as an artifact of interference among ψ(4160), ψ(4415), and nonresonant background [136]. The lower Y (4260) mass, with an asymmetric high-mass shoulder, was previously proposed based on the older data in the ¯ DD1 molecular model [137]. ππψ(2S) final state: Peaks in the effective mass of π+π−ψ(2S) states have been seen by Belle and BaBar around 4360 and 4660 MeV [138, 129, 139, 130] (Fig. 4). The former (“Y (4360)”) is roughly in the mass range expected for a charmonium 4S state, so the true 4S cc¯ amplitude might be shared between the Y (4360) and the ψ(4415), with the P + rest of the ψ(4415) wavefunction as a shallowly bound S-wave state of D1 (J = 1 ) and D∗ (J P = 1−). Alternatively, 4360 MeV is a plausible threshold for production of a D(0+)D¯ ∗(1−) pair. The latter (“Y (4660)”) could be associated with a peak at a + − + − nearby mass, about 4630 MeV, in e e → Λc Λc [132] (Fig. 4). The most precise data on the π+π−ψ(2S) channel in the lower-peak region was recently published by BESIII (Fig. 4), which found a significant evidence for a state at 4.21 GeV, perhaps the same one as observed in the π+π−J/ψ channel, and improved the Y (4360) mass determination to 4384 ± 4 MeV [128]. 1 ππhc final state: The hc(3525) is the lowest-lying (n = 1) n P1 charmonium level, first seen by the CLEO Collaboration [140]. It is curious that, as a spin-singlet level, it should

15 5 ʚ4160 ʛ ʚ3770 ʛ ʚ4040 ʛ ʚ4415 ʛ 4

͌ BES III 3 2 ͙͙ͮͯ → ͕ͦͣͧ͘͜͢ ̈́/

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ͯ 40 ͙ ͮ

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͙ 0 ͮ ͙ 

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͙ 0 0  ͮ ͙ 

[pb] 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 Karliner,Rosner,Skwarnicki 2017 ECM [GeV]

Figure 4: Measurements of cross-sections for e+e− annihilation: (top left inset) to hadrons expressed in units of R [125], (top row) to π+π−J/ψ [127, 115], (middle row) to π+π−ψ(2S) + − [128, 129, 130], (bottom row, left) to π π hc (black/red points are from two different + − energy scans by BESIII) [131], and (bottom row, right) to Λc Λc [132]. The displayed curves were fitted to the BESIII data (see Refs. [127, 128, 131]).

16 + − + − have been produced in e e → π π hc, as first observed by the CLEO Collaboration [141] and reported recently by the BESIII Collaboration [131]. Normally one would expect a virtual photon to produce a cc¯ spin-triplet state, so the process must be violating heavy- quark symmetry, perhaps via an intermediate open-flavor-pair state [113] in which the correlation between heavy-quark spins is lost. Two resonant structures are seen, one at 4218±5 MeV and a broader one at 4392±7 MeV (Fig. 4). The first is consistent with the BESIII observation in the π+π−J/ψ and π+π−ψ(2S) final states mentioned above, while the second could be an artifact of interference among ψ(4160), ψ(4415), and nonresonant background [136]. Open charm final states: A comprehensive analysis of the behavior of the cross section for production of open charm final states has been made in Ref. [142]. The analysis supports the identification, mentioned above, of the Y (4260) as mainly a molecular state ¯ + − ∗ ¯ ∗ + − ∗ ¯ ∗ of DD1(2420). The resonance line shapes for e e → D D and e e → Ds Ds can be satisfactorily explained with contributions from ψ(4040), ψ(4160), and ψ(4415), assuming suitable relative phases. Resonances Υ(5S) and Υ(6S): The behavior of R above BB¯ threshold exhibits two +10 bumps, called Υ(5S) and Υ(6S), with respective masses 10890 ± 3 and 10993−3 MeV [79]. An example of the shape of these bumps is given in Ref. [143]. Decay modes (∗) ¯(∗) common to both include B(s) B(s) , ππΥ(1S, 2S, 3S), and ππhb(1P, 2P ). As in the case of Y (4260) → ππhc, the latter class of decays violates heavy-quark symmetry and points to the role of open-flavor intermediate states [113]. The large decay widths for the transitions to ππΥ(nS) may be understood as enhancements of decay rates due to the intermediate states Zb(10610) and Zb(10650) [108]. Several other decay modes, including f0Υ(1S), ηΥ(1S, 2S), and π+π−Υ(1D), are reported for Υ(5S).

4.5 Other Exotic Meson Candidates Detected in B Decays ¯ ¯ + + The first evidence for an explicitly exotic charged QQud state, Zc(4430) → ψ(2S)π , ¯ + 0 − was claimed by the Belle collaboration in B → ψ(2S)π K decays (K = KS or K ) in + − ∓ ± 2007 [144], well before the other charged candidates were observed in e e → π Zb,c (see Sec. 4.3). This state has a vivid experimental history. It was first claimed as a narrow +35 ± peak (Γ = 45−18 MeV) in the invariant ψ(2S)π mass distribution [144], with parameters obtained by a na¨ıve fit to this distribution with ad hoc assumptions about the shape of the background from excited kaons, K∗ → Kπ+, dominating such B decays. This observation was soon questioned by the BaBar experiment [145]. In response, the Belle experiment published amplitude analyses with a realistic model of K∗ resonances, first performed on the Dalitz plane [146], later also including angular information from ψ(2S) → `+`− decays P + + + [147], which pointed to a significant J = 1 Zc(4430) → ψ(2S)π contribution, albeit +49 much broader than initially claimed (Γ = 200−58 MeV). Later, the LHCb collaboration confirmed the Belle results in a similar amplitude analysis performed on a much larger + sample of B decays [148], and demonstrated consistency of the Zc(4430) peak with a resonant hypothesis using an Argand diagram. They also demonstrated a need for other significant contributions than K∗0 → K−π+ to B¯0 → ψ(2S)π+K− decays without any assumptions about K∗ resonances, other than limiting their spin in the relevant low K−π+ mass region [149].

17 + + ¯0 The Belle collaboration claimed to have spotted Zc(4430) → J/ψπ in B → J/ψπ+K− decays, this time producing a dip in the ψ(2S)π+ mass distribution via inter- + 99 + + + ference with an even broader (Γ = 370−149 MeV) second 1 state, Zc(4200) → J/ψπ , + + [150]. There was also some indication for a second Zc → ψ(2S)π state around that mass with 0− or 1+ quantum numbers in the LHCb data [148]. + The Belle collaboration also reported evidence for charged χc1π resonances, the + + ¯0 + − Zc(4050) and Zc(4250) , in the amplitude analysis of B → χc1π K decays, but could not determine their quantum numbers [151]. BaBar saw an enhancement in the same + ∗ χc1π mass region, but suggested it could be a reflection of K resonances [152]. Without an amplitude analysis, their results do not contradict the Belle results. + As broad states, the charged Zc candidates are poor candidates for molecules of D and D¯ excitations. They have not been reported in prompt production at the Tevatron or LHC, thus also making poor candidates for tightly bound tetraquark states. It is − + ∓ ± remarkable that they have not been observed in the e e → π Zc reaction; and, vice + versa, the Zc states observed there have not been seen in B decays. This points to hadron-level forces responsible for these structures, perhaps via hadron rescattering in B decays, as such forces are expected to be sensitive to details of production mechanisms. Future high-statistics amplitude analyses of B decays in the upgraded LHCb and Belle experiments should shed more light on these effects. + The history of the X(4140) state has some parallels to the Zc(4430) saga. It was first +9.1 claimed in 2008 as a narrow peak (Γ = 11.7 −6.2 MeV) observed by the CDF collaboration in the invariant J/ψφ mass distribution from B+ → J/ψφK+ decays [153]. The existence of such a narrow, near-threshold state was questioned by the LHCb experiment [154]. The CMS experiment confirmed its existence, however, with somewhat larger width [155]. Later the LHCb experiment analyzed the biggest to date sample of B+ → J/ψφK+ decays, and performed the first amplitude analysis of this channel, thus providing more realistic subtraction of the B+ → J/ψK∗+, K∗+ → φK+ backgrounds [156, 157]. The LHCb data are consistent with a near-threshold J/ψφ resonance, however, with a much broader width +30 (Γ = 83 −25 MeV) than initially measured. LHCb determined its quantum numbers to be J PC = 1++. Since the 2011 update of the CDF analysis, there was a hint for a second X(4274) → J/ψφ state in the same B+ decay mode [158]. A second J/ψφ mass enhancement was visible in the CMS data, but at a higher mass [155]. The amplitude analysis by LHCb confirmed the X(4274) state with high statistical significance and determined its quantum numbers to be also 1++ [156, 157]. Two 0+ states at higher masses, X(4500) and X(4700), were also needed for a good description of the LHCb data. The D0 experiment presented an evidence for prompt production of X(4140) in pp¯ collisions at Tevatron [159]. It is puzzling why the X(4140) width observed in this inclusive measurements was narrow (Γ = 16±13 MeV) and why the X(4274), X(4500) and X(4700) were not observed. This observation awaits a confirmation. The Belle experiment, which was lacking statistics in the B+ → J/ψφK+ channel, looked for J/ψφ states in γγ collisions. They obtained evidence for a narrow X(4350) +18 state (Γ = 13−10 MeV) and saw no other J/ψφ mass peaks [160]. The X(4350) state awaits confirmation as well. The origin of the J/ψφ states, among which the X(4140) and X(4274) should be

18 considered experimentally established, is far from clear. Their masses do not fall into ¯ the mass intervals near the pairs of excitations of the Ds (Ds) mesons with matching quantum numbers for S-wave interactions, bound by η exchange [40]. Explanation of ± ∗∓ the X(4140) as related to a Ds Ds cusp [161, 157], relies on broadening this threshold effect via a poorly justified form factor. Tightly bound tetraquark models can account for a doublet of 1++ states only in an approach in which “good” (color antitriplet) and “bad” (color sextet) diquarks are allowed [162]. In the tetraquark model using only “good” diquarks, it was suggested that X(4274) is not a 1++ state but a superposition of two states with 0++ and 2++ [34]. However, such components of X(4274) are disfavored by more than 7 standard deviations by the LHCb analysis (Table 7 in Ref. [157]). It was 3 3 suggested that X(4274), X(4500) and X(4700) states are conventional 3 P1, 4 P0 and 3 3 5 P0 charmonium states, respectively [163]. However, no explanation of why 4 P1 and 3 5 P1 states would not be also visible in the J/ψφ decay mode was offered. None of the X → J/ψφ states observed in B+ → J/ψφK+ decays is seen in the J/ψω decay mode probed in B+ → J/ψωK+ decays (see Fig. 43 in Ref. [164]), suggesting that the ss¯ pair is among the constituents of these states. With no plausible theoretical interpretation of all four of them together, they may have different origins or be some complicated (∗) artifacts of rescaterring of Ds(J) meson-antimeson pairs. Future higher-statistics samples of B+ → J/ψφK+ decays may allow probing the nature of these structures in a less model-dependent way and shedding more light on their nature. A near-threshold enhancement in the J/ψω mass distribution in B → J/ψωK decays was first reported by Belle [165]. BaBar later resolved this structure into two mass peaks, identified with X(3872) → J/ψω decay (Sec. 4.2) and the state at 3919 ± 4 MeV with rather narrow width, Γ = 31 ± 11 MeV [87] (Fig. 1). Both Belle and BaBar observed a state at similar mass and width in γγ collisions [166]. It is commonly assumed, but not proven, that these mass structures are due to the same state as the one called X(3915), with 0++ or 2++ as likely quantum numbers. This state is too narrow to be a conventional charmonium triplet P -state (for a full discussion see Ref. [164]) at masses where decays to ¯ ∗ ¯ 3 DD and DD are allowed. It was recently proposed that mixing of the 2 P2 charmonium state with a molecular DD¯ ∗ or D∗D¯ ∗ component could be responsible for X(3915) [167].

4.6 Quarkonium-like Pentaquark Candidates A possibility of four quarks and one antiquark binding together was anticipated from the beginnings of the quark model [1], later reinforced by QCD, in which a diquark can effectively act as an antiquark, thus two diquarks and one antiquark can attract each other by the same means as three antiquarks do in an ordinary antibaryon. However, even today, we can’t directly predict from QCD if such bound states can live long enough to have any measurable effects. Pentaquarks made only out of up and down quarks lack useful experimental signatures to distinguish them from ordinary baryons. Pentaquarks with a flavored antiquark would decay strongly to a baryon and a flavored meson, a final state which cannot be produced in a decay of an ordinary baryon. While some pentaquark candidates of that type were claimed in the past experiments, none of them survived scrutiny of additional data (see Secs. 2.2,3.2). In 2015, the LHCb experiment observed a rather narrow (Γ ∼ 40 MeV) structure in

19 250

1000 m all mKp>2 GeV data Kp 200

150 total fit 800 100

background Events/(20 MeV) 50 Events/(15 MeV) 600 Pc(4450) 0 4 4.2 4.4 4.6 4.8 5

Pc(4380) 400 LHCb 200

0 4 4.2 4.4 4.6 4.8 5 mJ/ψp [GeV]

+ + Figure 5: Observation of the pentaquark candidates Pc(4450) and Pc(4380) decaying − to J/ψp in the amplitude analysis of Λb → J/ψpK decays by the LHCb collaboration. Adapted from Ref. [168].

− the J/ψp mass distribution in Λb → J/ψpK decays [168], as shown in Fig. 5. Since the heavy cc¯ pair in the J/ψ cannot be created during hadronization with rates which would lead to such observation, this structure makes for a convincing uudcc¯ candidate. Its statistical significance is much larger than any of the previous pentaquark candidates, thus this effect is not going to fade away with additional data. LHCb demonstrated at the 9σ level that the J/ψp mass peak cannot be due to reflections of excited Λ states decaying − ∗ to pK , with almost no assumptions about such Λ baryons, which dominate this Λb decay mode [169]. An amplitude analysis of these data, which used 13 well established Λ∗ resonances as a model for the pK− component, revealed that two J/ψp resonances + were needed for a reasonable description of data: the narrow Pc(4450) (Γ = 39 ± 20 + MeV) and the lighter and wider Pc(4380) (Γ = 205 ± 88 MeV). Both states had a very high statistical significance (12σ and 9σ, respectively), albeit depending on much stronger model assumptions [168]. The Dalitz plot pattern of their intensities implies they should have opposite parities. The spin combinations involving 3/2 and 5/2, in either order, were preferred. In addition to pentaquarks with uudcc¯ quarks bound together in one confining volume by color forces, also baryon-meson molecules bound by residual color forces, similar to those responsible for creation of nuclei, can have the same quark ¯ ∗ content. In fact, a ΣcD molecular state was predicted by Karliner and Rosner around the + P − Pc(4450) mass [39]. This model requires J = 3/2 and provides a natural explanation + for its narrow width. The pχc1 mass threshold coincides with the Pc(4450) mass [170]. Such a molecular state, or cusp, would require J P = 3/2+. Molecular bound states or + cusps don’t offer any explanation for the broad Pc(4380) state, nor can they lead to spin as high as 5/2 in this mass range. Rescattering of ordinary baryons and mesons,

20 via the so-called triangle anomaly, must happen in an S-wave to be pronounced, thus cannot account for spin 5/2 either [171, 172, 43]. The tightly bound pentaquark model + can generate such high spin for Pc(4450) via orbital angular momentum between quarks + [33] and can account for the wider Pc(4380) . So far, the rich mass spectrum necessarily resulting from such quark confinement has not been experimentally observed. It is also not clear why such a pentaquark state would be narrow, with the large phase-space available for J/ψp decays and the spatial proximity of c andc ¯. It was suggested that momentary separation of c andc ¯, followed by immediate hadronization, can be a result of a production mechanism pushing them in opposite directions [173]. Such a cartoon model is lacking predictive power, thus is difficult to confirm or dismiss. The LHCb Collaboration did not assign statistical or systematic significance to the determined quantum number preference. Therefore, it is premature to draw strong conclusions about possible interpretations of + the Pc states based on this preference. It is more than likely that the LHCb model of the Λ∗ states was incomplete, since about 60 Λ∗ states are predicted in the relevant mass range by the quark model [168], and in fact some of them were observed in various analyses of the KN scattering data, but were too model-dependent to earn labels of well- established states by the PDG [25]. Coupled channels, especially (Σπ)I=0, are likely to − make significant contributions as well. More Λb → J/ψpK data are already available to LHCb. It is hoped their improved amplitude analysis will shine more light on the nature of these J/ψp mass structures. − The LHCb analyzed also the Cabibbo-suppressed channel Λb → J/ψpπ [174]. With much fewer events, complications from many known pπ− resonances, and the possibility of − − an exotic contribution from the Zc(4200) → J/ψπ state, the results were inconclusive. + + The data are fully compatible with the Pc(4380) and Pc(4450) contributing to this final state at the expected level, but also compatible with no such contributions if the − Zc(4200) is allowed. + There have been no claims of spotting the Pc states in prompt production at LHC, which would have favored a tightly bound pentaquark model. Molecular or tightly bound J/ψp states should be reachable in photoproduction at JLab [175], where several experi- mental searches for them are under way.

5 BEYOND DETECTED STATES

QQQ¯Q¯: The question of whether there exist bound states of two heavy quarks Q = (c, b) and antiquarks Q¯ = (¯c, ¯b), distinct from a pair of quark-antiquark mesons, has been debated for more than forty years. It has drawn substantial interest recently [176, 177, 178].

Ref. [176] predicted M(Xccc¯c¯) = 6,192 ± 25 MeV and M(Xbb¯b¯b) = 18,826 ± 25 MeV, for the J PC = 0++ states involving charmed and bottom tetraquarks, respectively. Earlier predictions vary over a big range, with large error bars, cf. Table VII in Ref. [176]. A more recent compilation of predicted values of M(Xbb¯b¯b) − 2M(ηb) appears in Table I of Ref. [177]. The proximity of the predicted Xbb¯b¯b mass to 2M(ηb) = 18, 798 ± 5 MeV [25] and the size of the theoretical errors suggests that Xbb¯b¯b either decays strongly with a rather narrow width, or it is below the ηbηb threshold, in which case one expects final

21 states of hadrons from pairs of intermediate gluons, and of hadrons or leptons from pairs of intermediate virtual photons. Experimental search for these states in the relevant mass range is highly desirable. Searches in the four-lepton and `+`−BB¯ final states have been performed at the LHC [179]. These are devoted to the search for the standard-model Higgs boson decaying into two light pseudoscalars a, which then decay to such final states as µ+µ−, τ +τ −, and b¯b. These are ideal samples for the searches advocated here. ∗ Bottom analogues of Ds0(2317) and Ds1(2460): These BsJ states are the yet-to-be-discovered b-quark analogues of the very narrow P + DsJ states seen by BaBar, CLEO and Belle [58, 59, 60, 61] Ds0(2317) with J = 0 and P + P − ∗ m[Ds1(2460)] with J = 1 , conjectured to be the chiral partners of Ds, J = 0 and Ds , J P = 1−, respectively [71, 72]. A strong hint toward this conjecture is supplied by almost equal splitting between the states of opposite parity [25]: m[Ds0(2317)] − m[Ds]=349.4 ± ∗ 0.6 MeV≈ m[Ds1(2460)] − m[Ds ]= 347.3 ± 0.7 MeV≈ constituent mass of light quarks. Assuming approximately the same splitting in the bottom sector, one expects Bs0 at P + P + ∼ 5717 MeV with J = 0 and Bs1 at ∼ 5765 MeV with J = 1 . They are also predicted by a lattice calculation [180]. These states are likely to be observed at LHCb + − ¯∗ + − ¯ and might also be accessible at Belle II in e e → Bs0Bs and e e → Bs1Bs [181]. Stable bbu¯d¯ tetraquark: ++ Recently LHCb discovered the first doubly-charmed baryon Ξcc = ccu at 3621.40 ± 0.78 MeV [182], very close to the theoretical prediction 3627 ± 12 MeV in Ref. [183].3 In Ref. [184] the same theoretical approach was used to predict a doubly-bottom tetraquark T (bbu¯d¯) with J P =1+ at 10, 389 ± 12 MeV, 215 MeV below the B−B¯∗0 threshold and 170 MeV below threshold for decay to B−B¯0γ. Similar conclusions were obtained in Refs. [185]. The T (bbu¯d¯) is therefore stable under strong and electromagnetic (EM) interactions and can only decay weakly, the first exotic hadron with such a property. The predicted lifetime is τ(bbu¯d¯) ∼ 367 fs. The T (bbu¯d¯) tetraquark can decay through one of two channels: (a) The “standard process” bbu¯d¯ → cbu¯d¯+ W ∗−. Typical reactions include T (bbu¯d¯) → D0B¯0π−, D+B−π− and T (bbu¯d¯) → J/ψK−B¯0, J/ψ K¯ 0B−. In addition, there is a rare process where both b quarks decay into ccs¯ , T (bbu¯d¯) → J/ψJ/ψK−K¯ 0. The signature for events with two J/ψ ’s coming from the same secondary vertex might be sufficiently striking to make it worthwhile to look for such events against a large background. (b) The W -exchange process bd¯→ cu¯, involving either one of the two b quarks. The latter process can involve a two-body final state, e.g., T (bbu¯d¯) → D0B−. In contrast with T (bbu¯d¯), the mass of T (ccu¯d¯) with J P =1+ is predicted to be 3882±12 MeV, 7 MeV above the D0D∗+ threshold and 148 MeV above D0D+γ threshold. T (bcu¯d¯) with J P =0+ is predicted at 7134 ± 13 MeV, 11 MeV below the B¯0D0 threshold. The theoretical precision is not sufficient to determine whether bcu¯d¯ is actually above or below the threshold. It could manifest itself as a narrow resonance just at threshold. At this point it is interesting to point out an interesting pattern: the known candidates for hadronic molecules are hidden-flavor quarkonium-like states QQq¯ q¯, Q = b, c, q = u, d, while the stable tetraquark belongs to the open-flavor QQq¯q¯ category. There is a good

3We refer the reader to Refs. [182] and [183] for an extensive list of other predictions, most of which quote much greater uncertainties.

22 reason for this pattern. T (bbu¯d¯) is below two-meson threshold because the two heavy quarks are very close to each other ∼0.2 Fermi. They form a color antitriplet and attract each other very strongly. Consider a typical Coulomb + linear Cornell-like potential V (r) = −αs/r + σr. At ∼0.2 Fermi the heavy quarks probe the Coulomb, singular part of the potential, so the binding energy is very large, ∼280 MeV. But the tightly-bound (bb) sub-system is a color antitriplet, so it cannot disconnect from the two light antiquarks. Hence the tetraquark is bound vs. two heavy-light bq¯ mesons which lack the strong attraction between the two heavy quarks. The situation is completely different in bottomonium-like system (b¯bqq¯): the lowest energy configuration of the (b¯b) subsystem is a color singlet. So when b and ¯b get close, they decouple from the light quarks and form an ordinary bottomonium. In other words, in a (b¯bqq¯) system there is no possibility of utilizing the very strong attraction between b and ¯b without at the same time forcing the system to decay into quarkonium and pion(s). This is why exotic bottomonium-like states have a completely different structure – they are hadronic molecules of two heavy-light mesons bound by exchange of light hadrons. Such molecules have a mass which is much higher than their decay products: For example, Zb(10610) is ∼1 GeV above Υ(1S)π threshold. Nonetheless, they have a strikingly narrow width despite such a large phase space, e.g., Γ(Zb(10610)) ∼ 20 MeV [186]. The reason is that in order to decay into quarkonium and a pion the two heavy quarks must get very close to each other. In a large deuteron-like molecular state the probability for such a close encounter is quite small, analogous to the small probability for an electron to be inside the proton in the ground state of a hydrogen atom. Analogous comments apply to ccq¯q¯ states vs. ccq¯ q¯, states, with an important difference that mc/mb ∼ 1/3, so the substantial binding energy of the (cc) subsystem is nevertheless significantly smaller that in (bb) subsystem and therefore ccq¯q¯ is likely unbound with respect to two (cq¯) mesons.

6 SUMMARY AND OUTLOOK

The quark model has been highly successful in describing the spectroscopy of mesons and baryons as quark-antiquark (qq¯) and three-quark (qqq) systems, respectively. With the u, d, s quarks assigned the fractional charges 2/3, −1/3, −1/3, these are the simplest states with integral charges. However, the model also implied the existence of more complicated states with integral charges, such as qqq¯q¯ mesons and qqqqq¯ baryons [1]. Regarding quarks as fundamental triplets of a color SU(3) symmetry, the color-singlet states have integral charges. Why weren’t these “exotic” states seen? One signal of an exotic hadron is its “flavor” quantum numbers, calculated from the charge (2/3, −1/3, −1/3) and strangeness (0, 0, −1) of the u, d, s quarks. Thus, a meson with the quantum numbers of uus¯s¯, decaying to K+K+ or pΞ¯+, would be manifestly exotic, as its charge and strangeness could not be exhibited by any qq¯ state. Similarly, a baryon with the quantum numbers of uudds¯, decaying to K+n or K0p, would be manifestly exotic. Various models implied that exotic states of u, d, s quarks existed, but were not iden-

23 tifiable either because they did not possess exotic flavor quantum numbers or because they were too broad to be distinguishable from two-hadron continuum states. A model in which quarks were confined by a quantum-chromodynamics “bag” [26] predicted a qqq¯q¯ meson as light as a few hundred MeV but with large decay width to ππ. Narrower exotic mesons now known as f0 and a0 were expected with masses about a GeV, as seen, but their flavor quantum numbers are indistinguishable from those of qq¯0 states. In the bag model they possess an additional ss¯ pair in their wave functions, and thus are known as “crypto-exotic.” The application of quark-hadron duality to baryon-antibaryon scat- tering implied that t-channel exchanges of (non-exotic) qq¯ states were dual to (exotic) qqq¯q¯ states in the s channel [28]. Thus, one expected exotic states in baryon-antibaryon channels, such as ∆++n¯ and Λ¯pπ+. Their absence may be ascribed to their large decay widths. The situation has changed with the advent of the heavy charm (c) and bottom (b) quarks. A multitude of exotic hadrons with two or more heavy quarks have been seen, starting with the X(3872) [35], where the number in parentheses refers to the mass in MeV. Several mechanisms appear to be at work in these observations. In the case of the X(3872), one-pion-exchange between a charmed meson D and an anti-charmed meson D¯ ∗, binding them into a bound or virtual S-wave state, plays a crucial role. The spin J, parity P , and charge-conjugation eigenvalue C of the state are then expected to be J PC = 1++, as observed. The isospin splitting between neutral and charged charmed mesons ensures that the meson-antimeson component of the X wave function is mainly D0D¯ ∗0, so it is a mixture of isospins zero and one with quark content ccu¯ u¯. Its decay to J/ψγ and ψ(2S)γ implies that it has some cc¯ in its wave function. It would then be a mixture of a molecular state and the first radial excitation of the χc1(1P ) state. [The notation is that of Ref. [25]]. The name “tetraquark” conventionally refers to a state in which all quarks and an- tiquarks participate democratically in binding. For the X(3872) to be identified as a tetraquark, the grouping into a charmed meson-antimeson pair has to be ignored, and P there has to be a charged partner with the same J nearby in mass. The Zc(3900) would have been a possible candidate except that it has C = − instead of C = + [79]. In the absence of full isospin multiplets, one cannot yet identify many exotic hadrons as tetraquarks or pentaquarks. The bottom counterpart of the X(3872) has yet to be identified. It may participate in some mixing with a state thought to be the χb1(3P ) [102]. Because its constituent B and B¯∗ mesons enjoy little isospin splitting, its isospin is expected to be mainly zero, so it should decay to Υ(1S, 2S, 3S)ω, unlike X(3872) which decays both to J/ψω and J/ψρ [104]. Strong candidates for molecular states also exist in the bottom sector. The masses of ¯∗ ∗ ¯∗ Zb(10610) and Zb(10650) are very close to the BB and B B thresholds, respectively. They are seen not only in the Υ(1S, 2S, 3S)π channels (only charged pion for Zb(10650)), ± but also in the hb(1P, 2P )π channels, implying a violation of heavy-quark symmetry [113]. This is to be expected if the wave functions of the states are mainly meson- antimeson. The important role of pion exchange in creating these states is supported by the absence of states near BB¯ threshold. A number of exotic meson candidates with cc¯ accompanied by light quarks have been

24 seen in the 4–5 GeV mass range. Many of these cannot be associated with specific thresholds, and their tetraquark interpretation often awaits discovery of their isospin counterparts. In channels where pion exchange is not possible, the role of η exchange remains to be tested [40]. A prominent feature of charmed-pair production by e+e− collisions is its rapid drop just above a center-of-mass energy of 4.2 GeV, recalling similar behavior in the ππ I = J = 0 channel just around KK¯ threshold. In the charm case the behavior is likely to be correlated with the threshold for D(2420)D¯ production, which is the lowest-lying charmed pair which can be produced in an S wave. It illustrates the importance of S- wave thresholds, which appear in a wide variety of cases in particle physics and elsewhere [124, 48]. After a couple of false starts (by others) in searches for qqqqq¯ states, the LHCb ex- − periment has observed two in the J/ψp channel, produced in the decay Λb → J/ψK p: a narrow one around 4450 MeV and a much broader one around 4380 MeV, with opposite ¯ ∗ parities and preference for 3/2 and 5/2 spins, in either order. A ΣcD molecule with properties consistent with the Pc(4450) state has been suggested [39], but a molecular interpretation of the lower, broader, state is elusive. A genuine pentaquark interpretation would imply the states are accompanied by numerous isospin partners, not yet observed. A general feature of exotics with two or more heavy quarks is the reduction in kinetic energy afforded by their large masses. This, together with their shorter Compton wave- length leading to deeper binding, implies that states incorporating those heavy quarks may be deeply enough bound to overcome the tendency to “fall apart” exhibited by ex- otic states composed only of u, d, s quarks. An extreme example of this is the prediction of a bound bbu¯d¯ state [184, 185], supported by methods used in the successful prediction of the mass of a baryon containing two charm quarks [183]. Looking back at the experimental developments in hadron spectroscopy in the new millennium: heavy quarks have done it again! After converting us into firm believers of the quark model in the seventies, heavy quark systems have more recently taught us a new lesson: not all hadronic states are the minimal quark combinations. In addition to qq¯ mesons, four-quark qqq¯q¯ configurations become important, especially near and above the qq¯ plus qq¯ meson thresholds. Similarly, not all baryons are qqq states; qqqQQ¯ con- figurations also play a role. Theoretical disputes rage on, if the observed multiquark configurations are tightly bound tetra- and penta-quarks, or loosely bound meson-meson and baryon-meson molecules. In our opinion, the case for the latter is stronger. It is also beyond any dispute that baryon-baryon molecules exist and have been known for a long time as nuclei. This does not imply that every multiquark system must be loosely bound. In fact, the models which work well for doubly-charmed baryons also predict a stable bbu¯d¯ tightly bound tetraquark. What does the future hold for exotic multiquark mesons and baryons? As mentioned, the photoproduction of J/ψp resonances is possible at JLAB. Production of charmonium- like states is envisioned at PANDA. We are likely to be surprised by more charmonium-like exotics from Belle II and LHCb. After its upgrades, the LHCb may have a shot at the bbu¯d¯ tetraquark. We are looking forward to these developments!

25 DISCLOSURE STATEMENT

The authors are not aware of any affiliations, memberships, funding, or financial holdings that might be perceived as affecting the objectivity of this review.

ACKNOWLEDGMENTS

This work was supported by the National Science Foundation (USA) Award Number 1507572.

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